For locally convex pleated surfaces, Mess noticed that the left and right projections are earthquakes (see e.g. [1]}). Moreover (see [2]}) if \(S\)  is a complete locally convex pleated surface in \(\mathbb {ADS}^3\) , then the left (resp. right) projection \(\Pi _L\)  (resp. \(\Pi _R\) ) is an earthquake map, which extends the left (resp. right) projection of \(\partial S\)  to \(\partial P_0\) .
